

A070932


Possible number of units in a finite (commutative or noncommutative) ring.


3



0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 36, 40, 42, 44, 45, 46, 48, 49, 52, 54, 56, 58, 60, 62, 63, 64, 66, 70, 72, 78, 80, 81, 82, 84, 88, 90, 92, 93, 96, 98, 100
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OFFSET

1,3


COMMENTS

This is a list of the numbers of units in R where R ranges over all finite commutative or noncommutative rings.
By considering the ring Z_n and the finite fields GF(q) this sequence contains the values of the Euler function phi(n) (A000010) and prime powers  1 (A181062). By taking direct product of rings, if n and m belong to the sequence then so does m*n.
Eric M. Rains has shown that these rules generate all terms of this sequence. More precisely, he shows this sequence (with 0 removed) is the multiplicative monoid generated by all numbers of the form q^nq^{n1} for n >= 1 and q a prime power (see Rains link).
Since the number of units of F_q[X]/(X^n) is q^n  q^(n1), restricting to finite commutative rings gives the same sequence. A296241, which is a proper supersequence, allows the ring R to be infinite.  Jianing Song, Dec 24 2021


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. M. Rains, Comments on A070932


MATHEMATICA

max = 100; A000010 = EulerPhi[ Range[2*max]] // Union // Select[#, # <= max &] &; A181062 = Select[ Range[max], Length[ FactorInteger[#]] == 1 &]  1; FixedPoint[ Select[ Outer[ Times, #, # ] // Flatten // Union, # <= max &] &, Union[A000010, A181062] ] (* JeanFrançois Alcover, Sep 10 2013 *)


PROG

(PARI) list(lim)=my(P=1, q, v, u=List()); forprime(p=2, default(primelimit), if(eulerphi(P*=p)>=lim, q=p; break)); v=vecsort(vector(P/q*lim\eulerphi(P/q), k, eulerphi(k)), , 8); v=select(n>n<=lim, v); forprime(p=2, sqrtint(lim\1+1), P=p; while((P*=p) <= lim+1, listput(u, P1))); v=vecsort(concat(v, Vec(u)), , 8); u=List([0]); while(#u, v=vecsort(concat(v, Vec(u)), , 8); u=List(); for(i=3, #v, for(j=i, #v, P=v[i]*v[j]; if(P>lim, break); if(!vecsearch(v, P), listput(u, P))))); v \\ Charles R Greathouse IV, Jan 08 2013


CROSSREFS

Cf. A000010, A002202, A000252, A000961, A181062, A221178.
A000252 is a subsequence.
A282572 is the subsequence of odd terms.
Proper subsequence of A296241.
The main entries concerned with the enumeration of rings are A027623, A037234, A037291, A037289, A038538, A186116.
Sequence in context: A238369 A296858 A296241 * A161577 A093686 A325031
Adjacent sequences: A070929 A070930 A070931 * A070933 A070934 A070935


KEYWORD

nonn,nice


AUTHOR

Sharon Sela (sharonsela(AT)hotmail.com), May 24 2002


EXTENSIONS

Entry revised by N. J. A. Sloane, Jan 06 2013, Jan 08 2013
Definition clarified by Jianing Song, Dec 24 2021


STATUS

approved



